Molecular Mechanics - Force Field Method

1. Applied Statistical Mechanics Lecture Note - 6 Molecular Mechanics - Force Field Method 고려대학교 화공생명공학과 강정원

2. Introduction • Force Field Method vs. Electronic Structure Method • Force field method : based on Molecular Mechanics • Electronic structure method : based on Quantum Mechanics

3. Force Field Method • Problem • Calculating energy for given structure • Finding stable geometry of molecules • Energy optimum of saddle point • Molecules are modeled as atoms held together with bonds • “Ball and spring model “ • Bypassing the electronic Schrödinger equation • Quantum effects of nuclear motions are neglected • The atom is treated by classical mechanics  Newton’s second law of motion

4. Force Field Methods • Validation of FF methods • Molecules tend to be composed of units which are structurally similar in different molecules • Ex) C-H bond • bond length : 1.06 – 1.10 A • stretch vibrations : 2900 – 3300 /cm • Heat of formation for CH3 – (CH2)n – CH3 molecules • Almost straight line when plotted against n • Molecules are composed of structural units  “Functional groups”

5. Example MM2 atom types

6. The Force Field Energy • Expressed as a sum of terms • Estr: The stretch energy • Ebend: The bending energy • Etor: The torsion energy • Evdw : The van der Waals energy • Eel : The electro static energy • Ecross : coupling between the first three terms Bonded interactions Nonbonded atom-atom interaction

7. The stretch energy A B • Estr : The energy function for stretching a bond between tow atom type A and B • Equilibrium bond length  Minimum energy • Taylor series expansion around equilibrium bond length Set to 0 0 at minimum energy Simplest form : Harmonic Oscillator

8. The stretch energy • The harmonic form is the simplest possible form • When the bond is stretched to longer r , the predicted energy is not realistic • Polynomial expansion • More parameters • The limiting behavior is not correct for some cases ( 3rd order, 5th order,…) • Special care needed for optimization (negative energy for long distance )

9. The stretch energy • The Morse Potential • D: Dissociation energy • Accurate actual behavior • Problem • More computation time evaluating exponential term • Starting from poor geometry, slow convergence • Popular method : nth order expansion of the Morse Potential

10. The Bending Energy • Ebend : The energy required for bending an angle formed by three atoms A-B-C • Harmonic Approximation • Improvement can be observed when more terms are included • Adjusting higher order term to fixed fraction • For most applications, simple harmonic approximation is quite adequate • MM3 force field : 6th term q

11. The bending energy • Angels where the central atom is di- or tri- valent (ethers, alcohols, sulfiteds, amines), represents a special problem • an angle of 180 degree  energy maximum • at least order of three • Refinement over a simple harmonic potential clearly improve the overall performance. • They have little advantage in the chemically important region ( 10 kcal/mol above minimum)

12. The out-of-plane bending energy • sp2-hybridized atoms (ABCD) • there is a significant energy penalty associated with making the center pyramidal • ABD, ABD, CBD angle distortion should reflect the energy cost associated with pyramidization

13. The torsion energy • Angle of rotation around B-C bond for four atoms sequence A-B-C-D • Difference between stretch and bending energy • The energy function must be periodic win the angle w • The cost of energy for distortion is often low • Large deviation from minimum can occur • Fourier series expansion

14. The torsion energy • Depending on the situation some of Vn terms are set to 0 • n=1 : periodic by 360 degree • n=2 : periodic by 170 degree • n=3 : periodic by 120 degree • Ethane : three minima and three maxima • n = 3,6,9, … can have Vn

15. + + - - The van der Waals energy • Evdw : energy describing the repulsion and attraction between atoms : non-bonded energy • Interaction energy not related to electrostatic energy due to atomic charges • Repulsion and attraction • Small distance, very repulsive  overlap of electron cloud • Intermediate distance, slight attraction  electron correlation • motion of electrons create temporarily induced dipole moment Repulsion Attraction

16. + + + - - - Van der Waals Attraction • 1930, London • “Dispersion” or “London” force • Correlation of electronic fluctations • Explained attraction as induced dipole interaction + -

17. Van der Waals Repulsion • Overlap of electron cloud • Theory provide little guidance on the form of the model • Two popular treatment • Inverse power • Typically n = 9 -12 • Exponential • Two parameters A, B

18. Van der Waals Energy • Repulsion + Atrraction gives two model • Lennard-Jones potential • Exp-6 potential • Also known as “Buckingham” or “Hill” type potential

19. Comparison Morse Potential Problems Inversion Overestimating repulsion

20. Why LJ potential is preferred ? • Multiplications are much faster than exponential calculation • Parameters are meaningful than the other models • Diatomic parameters

21. The electrostatic Energy • Iternal distribution of electrons • positive and negative part of molecule • long range force than van der Waals • Two modeling approaches • Point charges • Bond Dipole Description

22. Point charge method • Assign Columbic charges to several points of molecules • Total charge sum to charges on the molecule • Atomic charges are treated as fitting parameters • Obtained from electrostatic potential calculated by electronic structure method (QM)

23. Bond Dipole description • Interaction between two dipole • MM2 and MM3 uses bond dipole description • Point charge vs. Bond Dipole model • There is little difference if properly parameterized • The atomic charge model is easier to parameterize by fitting an electronic wave function  preferred by almost all force field

24. 1 2 3 Multibody interaction • Unlike van der Waals interaction, the three body interaction is quite significant for polar species • Two method • Explicit multibody interaction • Axilrod – Teller • Atom Polarization • Electrostatic interaction = (Intrisic contribution) + (dipolar term arising from the other atomic charges) • Solved iterative self-consistent calculation

25. Cross terms • Bonds, angles and torsions are not isolated • They couple with one another • Example • Stretch/bend coupling • Stretch/stretch coupling • Bend/bend coupling • Stretch/torsion coupling • Bend/torsion coupling • Bend/torsion/bend coupling …

26. Small rings • Small rings present a problem • their equilibrium angles are very different from those of their acyclic cousins • Methods • Assign new atom types • Adding sufficient parameters in cross terms

27. Conjugated systems • Butadiene (C=C-C=C ) • Same set of parameters are used for all carbon atoms • Bond length of terminal and central bonds are different ( 1.35 A and 1.47 A) • Delocalization of pi-electrons in the conjugated system • Approaches • Identifying bond combination and use specialized parameters • Perform simple electronic structure calculation • Implemented in MM2 / MM3 (MMP2 and MMP3) • Electronic structure calculation method (Pariser-Pople-Parr (PPP) type) : Extended Hückel calculation • Requires additional second level of iteration in geometry optimization If the system of interest contains conjugation, a FF which uses the parameter replacement is chosen, the user should check that the proper bond length and reasonable rotation barrier !

28. Comparing Energies of Different Molecules • The numerical value of force-field energy has no meaning ! • Zero point energy has been chosen for convenienece • It is inconsequential for comparing energies of different conformation • EFF : “ steric energy” • Heat of formation • Bond dissociation energy for each bond type • To achieve better fit, parameters may also be assigned to larger units (groups : CH3- ,…) • MM2/MM3 attemped to parameterize heat of formation • Other force fields are only concerned with producing geometries

29. Force Field Parameterization • Numerical Values of parameters • Example : MM2 (71 atom types) • For one parameters at least 3-4 independent data are required • Require order of 107 independent experimental data  Impossible • Rely on electronic structure calculation (“Class II” force field)

30. Force Field Parameterization • There are large number of possible compounds for which there are no parameters • The situation is not as bad as it would appear • Although about 0.2 % of possible torsional constants have been parameterized, • About 20 % of 15 million known compound can be modeled by MM2

31. Universal Force Field • Many atom type, lack of sufficient reference data  Development of Universal Force Field (UFF) • Derive di-, tri-, tetra- atomic parameters from atomic constant (Reduced parameter form) • Atomic properties : atom radii, ionization potential, electronegativity, polarizability, … • In principle, capable of covering molecules composed of elecments from the whole periodic table • Less accurate result presently … likely to be improved …

32. Force Fields…